This will be my first pathfinder game and I'm building a human fighter focusing on ranged attacks. I'm not really familiar with 3.5 or PF items, and was wondering if there are items I absolutely must have at some point in my campaign.

Also, I'm still debating between the archer archetype and sticking with vanilla fighter. I'm not too familiar with the dex cap and how realistic it might be that I exceed +6 dex bonus (I'm guessing I want to aim for mithril?)

2 Answers
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This list from Rogue Eidolon's Guide to Fighters should come in handy. The guide has them rated, and it's always worth having a proper build in mind so you know what to ask your DM for. But if you need a generic "wishlist of stuff that a master archer fighter would want" take a look below.

Armour:

Celestial Armor: Master Archer’s armour of choice. Even that Dex-happy build will have to stretch to get the 34 Dex that hits the Max Dex on this beauty.

Weapons, When it comes to a Fighter, the weapon makes the warrior. Thus, weapons are in their appropriate sections for each build.

Speed: Another Sattack per round is superb--if you have a Haste-happy Wizard in your group, though, pass on this.

Seeking: Seeking is an amazing enhancement for the Master Archer (and even for a backup weapon’s arrows for everyone else). Even after you have Improved Precise Shot, it's still very useful. If you're trying to save on the cost of your bow, though, consider using Seeking Arrows--it'll be clear when you need to use them (or maybe it will be rather hazy or blurry, but you get my point).

Energy: 1d6 extra damage is nothing to sneeze at, but beware of energy resistance. Make sure you have some way to get actual enhancement bonuses on your bow before adding these, since hitting at this point is crucial.

Rings:

Protection: More AC is good. More AC to apply to Touch and Flat-Footed both? That's gold.

Freedom of Movement: Your CMD should be fine, but some enemies have a ridiculous CMB, so having this never hurts.

Force Shield: Cheaper and better than upgrading an Amulet of Natural Armor from +2 to +3. Only buy this if you don’t have some other way to get a Shield bonus.

Wondrous:

Handy Haversack: It's a staple for a reason--carry your stuff and always find the item you want on top!

Cloak of Resistance: Get this up as high as you can to bolster your saves. It's relatively cheap, too.

Belt of Physical Stats: Get Strength first then Dex, unless you’re an Archer, then switch.

Amulet of Natural Armor: More AC? Always useful. Get Ring of Protection first if there's a tie.

Ioun Stone, Dusty Rose Prism: +1 to AC is always good. Get this before upgrading the Amulet of Natural Armor and its friends to +2.
Stone of Good Luck: It's a bit expensive, but it gives +1 to all your saves and skills and stacks with pretty much everything. If you ever start considering upping your +2 Wis headband to +4, I recommend waiting and getting this instead.

Ioun Stone, Pale Green Prism: Stone of Good Luck's more powerful older brother, it's more costly too. If you have a Bard or Bracers of Archery, it's not as impressive, but if not, the 10000 extra is clearly worth it later on when you can afford this for the +1 to hit.

Circlet of Persuasion: If you want to use Intimidate (or UMD), this is the item for you. It cheaply mimics having 6 higher Charisma.

Boots of Striding and Springing: Movement speed is very useful, especially before Armor Training speeds you back up. Get this unless you need Boots of Speed more.

Boots of Speed:Probably a better idea than a Speed weapon if you don't have a Wizard to Haste you, since it won't make your weapon's cost shoot up for other abilities. The limited rounds per day are its only weakness.

Goggles of Night: Switch these on when you need to see in the dark, unless you have Darkvision already. But, this is less useful for you than it is for many other classes.

Eyes of the Eagle: Hear the door creak open and act on the surprise round with these handy lenses that double as a hearing aid for some reason.

Ioun Stone, Dark Blue Rhomboid: Gaining Alertness gives you eventually +6 to Perception and Sense Motive, which is even better than Eyes of the Eagle, though it costs 4x as much, in part due to being slotless. And of course, its untyped, so it stacks with everything.

Carpet of Flying and its flying friends: Get Carpet of Flying, Wings of Flying, etc eventually so you can actually beat flying archers, particularly if you aren't an archer.

Bracers of Archery: This is very important to archers and nearly useless to everyone else--note that it doesn't stack with Bard's Inspire Courage, making it next to worthless if you have a Bard.

Headband of Mental Stats: This just isn't a very efficient way to get the Will save and Perception bonuses you probably want from the Wis boost. The Int choice is pretty good if it gets you a useful skill--that may even be how you pick up something exotic like Use Magic Device. Avoid Cha like the plague though--far better to just grab a Circlet of Persuasion.

Boots of Levitation / Slippers of Spider Climbing: Good for lower levels before you can afford to fly.

Get a Bow

Without a bow you're not an archer but a target. Bows are extremely vulnerable to sunder attempts. Dedicated archers, in particular, are extremely vulnerable to sunder attempts even from folks without any improvements to sunder attempts because most archers don't threaten their areas--and therefore can't make attacks of opportunity--while their bows are readied. So you get a bow made from special materials; I suggest living steel (the metal not the RPG). There doesn't appear to be an immediate equal to D&D 3.5's elvencraft (which simply let a bow function as a quarterstaff), so you're stuck with spiked armor if you don't want to drop your bow to stab a nearby orc. Then get a locked gauntlet or a weapon cord or both. Alternately, and embarrassingly, the Improved Unarmed Strike feat is a thing if you don't want to be all prickly.

If the campaign lasts for any length of time that enables your character to develop a reputation, some smartass monster's going to start with your bow and then carve you up when it's gone. So make sure it's never gone.

Get Arrows

A bow without arrows is a just a bad fishing pole. You're going to need a lot of arrows. And, unlike the swordsman's sword, all your arrows that hit and half that miss are destroyed. This can be expensive. You can waste your time chasing the ability to make your own arrows or take the Leadership feat and have your followers make them for you (they probably won't be very merry at that prospect, though), but no matter what you're going to need arrows for a variety of situations. There are a lot of different arrows. It's worth getting the basics: standard, bludgeoning, and slashing; and adamantine, cold iron, and silver. You probably won't be able to carry every permutation of these, but you should have 2 combat rounds of ammunition of each of the latter 5 and many, many more standard arrows for goofing off against typical foes and shooting apples from children's heads.

Get Perception

Your biggest advantage over everyone is the range of your weapon. The battle map just isn't big enough to handle the ranges at which you're attacking. You need to control encounter distance, and you do that by having the biggest, fattest, stupid-large Perception bonus you can possibly attain. So get Perception as a class skill, keep it maximized, devote obscene resources to making it bigger, and don't go blind. Anything that improves or protects your vision is solid gold.

The most common type of Perception bonus is the competence bonus. You get this from bracers of the falcon's aim, which if your DM doesn't immediately ban means you can expect the campaign to be extremely challenging. It's dirt cheap, in essence grants the Improved Critical feat for your bow, and grants a +3 competence bonus to Perception checks. You can get a permanent +8 racial bonus to Perception checks by chugging 20,000 gp of demon senses; it's worth the red ears or forked tongue. You can get an insight bonus to Perception checks via the fan-coverted spell pattern recognition. The high-level spell eaglesoul grants a +2 morale bonus to Perception checks versus evil creatures. You can get at least a +10 enhancement bonus to Perception checks with the spell acute senses. In other words, like most of Pathfinder, spellcasting rocks, and being without the Use Magic Device skill means you're boned--get the Use Magic Device skill and buy wands and staffs of spells that aren't in magic items.

There's one item--the third-party major artifact Tsystalor's breastplate--that renders the wearer immune to blindness. In D&D 3.5 this was accomplished by the 3rd-level spell vision of the omniscient eye, but apparently such an effect is far more valuable in Pathfinder. Follow rumors about and go on quests for this item. Almost nothing hurts the archer more than blindness; there are ways around it so you don't have to see, but always being able to see is always much better. The only other way is the 20th-level starsoul extraordinary ability from the sorcerous starsoul bloodline, and that's way too long to wait and way too weird for an archer.

You'll eventually want an item of true seeing (that's not Tsystalor's crazy breastplate). Of course you want truesight goggles--everyone does--, but until you can, the +1 weapon special ability seeking functions as a sort-of workaround.

Get Dispel Magic

A surprising number of spells baffle archery completely. Most of these are spells based around high winds, and usually cast by druids. These spells mean you lose. Heck, the solid fog spell is low-level enough that you could actually see it happen, and you're not using archery in it or through it. Like, at all. So you need to make it go away. If you've a spellcaster on your side that might be his job but the +1 weapon special ability dispelling on a few arrows is required after a certain point (that point, by the way, is as soon as you can afford +1 dispelling arrows).

Notes

Although single-target and a competence bonus, the spell hunter's eye grants a +20 to Perception checks and has a 1-minute-per-level duration; that's enough time to cast it, run away, and snipe a fool until he's dead.

I like the +2 weapon special ability endless ammunition, but almost as good is the +1 weapon special ability conserving, which makes it so your arrows are never wasted; combine with the +4 weapon special ability second chance and your attacks are rarely wasted.

The +4 weapon special ability nimble shot probably isn't worth it. +4 is a lot of resources to devote to not making a 5-ft. step.